LMFDB - Embedded newform 1584.3.i.c.881.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1584,3,Mod(881,1584)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1584, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1584.881"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1584 = 2^{4} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1584.i (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,16,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$43.1608738747$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 17x^{6} + 22x^{5} + 62x^{4} + 862x^{3} + 1045x^{2} + 5238x + 13491 $ x^8 - 2x^7 + 17x^6 + 22x^5 + 62x^4 + 862x^3 + 1045x^2 + 5238*x + 13491
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 396)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.1
Root			$2.95745 + 2.85804i$ of defining polynomial
Character	$\chi$	$=$	1584.881
Dual form			1584.3.i.c.881.8

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-8.36495i q^{5} +8.08375 q^{7} -3.31662i q^{11} -19.8173 q^{13} +30.8543i q^{17} -13.9261 q^{19} -24.1582i q^{23} -44.9723 q^{25} -8.62139i q^{29} -46.9816 q^{31} -67.6201i q^{35} -58.1398 q^{37} +12.6279i q^{41} +65.2203 q^{43} +57.8586i q^{47} +16.3470 q^{49} -58.8268i q^{53} -27.7434 q^{55} +77.0781i q^{59} -71.4716 q^{61} +165.771i q^{65} +100.111 q^{67} -24.6395i q^{71} -38.4829 q^{73} -26.8108i q^{77} +10.7777 q^{79} -50.4834i q^{83} +258.095 q^{85} +59.8093i q^{89} -160.198 q^{91} +116.491i q^{95} +44.6662 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{7} - 8 q^{13} - 40 q^{19} - 48 q^{25} - 40 q^{31} - 56 q^{37} - 24 q^{43} + 96 q^{49} + 120 q^{61} + 272 q^{67} - 256 q^{73} - 32 q^{79} + 432 q^{85} + 96 q^{91} - 24 q^{97}+O(q^{100}) $ 8 * q + 16 * q^7 - 8 * q^13 - 40 * q^19 - 48 * q^25 - 40 * q^31 - 56 * q^37 - 24 * q^43 + 96 * q^49 + 120 * q^61 + 272 * q^67 - 256 * q^73 - 32 * q^79 + 432 * q^85 + 96 * q^91 - 24 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1584\mathbb{Z}\right)^\times$.

$n$	$145$	$353$	$991$	$1189$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 8.36495i	− 1.67299i	−0.547975	−	0.836495i	$-0.684601\pi$
$5$	− 8.36495i	− 1.67299i	0.547975	−	0.836495i	$-0.315399\pi$
$6$	0	0
$7$	8.08375	1.15482	0.577410	−	0.816454i	$-0.304063\pi$
$7$	8.08375	1.15482	0.577410	+	0.816454i	$0.304063\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 3.31662i	− 0.301511i
$11$	− 3.31662i	− 0.301511i
$12$	0	0
$13$	−19.8173	−1.52441	−0.762205	−	0.647336i	$-0.775883\pi$
$13$	−19.8173	−1.52441	−0.762205	+	0.647336i	$0.775883\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	30.8543i	1.81496i	0.420093	+	0.907481i	$0.361997\pi$
$17$	30.8543i	1.81496i	−0.420093	+	0.907481i	$0.638003\pi$
$18$	0	0
$19$	−13.9261	−0.732951	−0.366475	−	0.930428i	$-0.619436\pi$
$19$	−13.9261	−0.732951	−0.366475	+	0.930428i	$0.619436\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 24.1582i	− 1.05036i	−0.850993	−	0.525178i	$-0.823999\pi$
$23$	− 24.1582i	− 1.05036i	0.850993	−	0.525178i	$-0.176001\pi$
$24$	0	0
$25$	−44.9723	−1.79889
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 8.62139i	− 0.297289i	−0.988891	−	0.148645i	$-0.952509\pi$
$29$	− 8.62139i	− 0.297289i	0.988891	−	0.148645i	$-0.0474911\pi$
$30$	0	0
$31$	−46.9816	−1.51554	−0.757768	−	0.652524i	$-0.773710\pi$
$31$	−46.9816	−1.51554	−0.757768	+	0.652524i	$0.773710\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 67.6201i	− 1.93200i
$36$	0	0
$37$	−58.1398	−1.57135	−0.785673	−	0.618642i	$-0.787683\pi$
$37$	−58.1398	−1.57135	−0.785673	+	0.618642i	$0.787683\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	12.6279i	0.307998i	0.988071	+	0.153999i	$0.0492152\pi$
$41$	12.6279i	0.307998i	−0.988071	+	0.153999i	$0.950785\pi$
$42$	0	0
$43$	65.2203	1.51675	0.758376	−	0.651817i	$-0.225993\pi$
$43$	65.2203	1.51675	0.758376	+	0.651817i	$0.225993\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	57.8586i	1.23103i	0.788123	+	0.615517i	$0.211053\pi$
$47$	57.8586i	1.23103i	−0.788123	+	0.615517i	$0.788947\pi$
$48$	0	0
$49$	16.3470	0.333612
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 58.8268i	− 1.10994i	−0.831870	−	0.554970i	$-0.812730\pi$
$53$	− 58.8268i	− 1.10994i	0.831870	−	0.554970i	$-0.187270\pi$
$54$	0	0
$55$	−27.7434	−0.504425
$56$	0	0
$57$	0	0
$58$	0	0
$59$	77.0781i	1.30641i	0.757181	+	0.653205i	$0.226576\pi$
$59$	77.0781i	1.30641i	−0.757181	+	0.653205i	$0.773424\pi$
$60$	0	0
$61$	−71.4716	−1.17167	−0.585833	−	0.810432i	$-0.699232\pi$
$61$	−71.4716	−1.17167	−0.585833	+	0.810432i	$0.699232\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	165.771i	2.55032i
$66$	0	0
$67$	100.111	1.49420	0.747098	−	0.664714i	$-0.231447\pi$
$67$	100.111	1.49420	0.747098	+	0.664714i	$0.231447\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 24.6395i	− 0.347035i	−0.984831	−	0.173518i	$-0.944487\pi$
$71$	− 24.6395i	− 0.347035i	0.984831	−	0.173518i	$-0.0555134\pi$
$72$	0	0
$73$	−38.4829	−0.527162	−0.263581	−	0.964637i	$-0.584904\pi$
$73$	−38.4829	−0.527162	−0.263581	+	0.964637i	$0.584904\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 26.8108i	− 0.348192i
$78$	0	0
$79$	10.7777	0.136426	0.0682131	−	0.997671i	$-0.478270\pi$
$79$	10.7777	0.136426	0.0682131	+	0.997671i	$0.478270\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 50.4834i	− 0.608233i	−0.952635	−	0.304117i	$-0.901639\pi$
$83$	− 50.4834i	− 0.608233i	0.952635	−	0.304117i	$-0.0983613\pi$
$84$	0	0
$85$	258.095	3.03641
$86$	0	0
$87$	0	0
$88$	0	0
$89$	59.8093i	0.672015i	0.941859	+	0.336008i	$0.109077\pi$
$89$	59.8093i	0.672015i	−0.941859	+	0.336008i	$0.890923\pi$
$90$	0	0
$91$	−160.198	−1.76042
$92$	0	0
$93$	0	0
$94$	0	0
$95$	116.491i	1.22622i
$96$	0	0
$97$	44.6662	0.460477	0.230238	−	0.973134i	$-0.426049\pi$
$97$	44.6662	0.460477	0.230238	+	0.973134i	$0.426049\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 104.218i	− 1.03186i	−0.856630	−	0.515932i	$-0.827446\pi$
$101$	− 104.218i	− 1.03186i	0.856630	−	0.515932i	$-0.172554\pi$
$102$	0	0
$103$	−28.0702	−0.272526	−0.136263	−	0.990673i	$-0.543509\pi$
$103$	−28.0702	−0.272526	−0.136263	+	0.990673i	$0.543509\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 14.0098i	− 0.130933i	−0.997855	−	0.0654665i	$-0.979146\pi$
$107$	− 14.0098i	− 0.130933i	0.997855	−	0.0654665i	$-0.0208535\pi$
$108$	0	0
$109$	−69.2977	−0.635758	−0.317879	−	0.948131i	$-0.602971\pi$
$109$	−69.2977	−0.635758	−0.317879	+	0.948131i	$0.602971\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0.214655i	0.00189960i	1.00000		0.000949801i	$0.000302331\pi$
$113$	0.214655i	0.00189960i	−1.00000		0.000949801i	$0.999698\pi$
$114$	0	0
$115$	−202.082	−1.75723
$116$	0	0
$117$	0	0
$118$	0	0
$119$	249.419i	2.09596i
$120$	0	0
$121$	−11.0000	−0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	167.067i	1.33654i
$126$	0	0
$127$	−157.887	−1.24320	−0.621602	−	0.783333i	$-0.713518\pi$
$127$	−157.887	−1.24320	−0.621602	+	0.783333i	$0.713518\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	94.3295i	0.720073i	0.932938	+	0.360036i	$0.117236\pi$
$131$	94.3295i	0.720073i	−0.932938	+	0.360036i	$0.882764\pi$
$132$	0	0
$133$	−112.575	−0.846427
$134$	0	0
$135$	0	0
$136$	0	0
$137$	36.9603i	0.269783i	0.990860	+	0.134891i	$0.0430686\pi$
$137$	36.9603i	0.269783i	−0.990860	+	0.134891i	$0.956931\pi$
$138$	0	0
$139$	92.1371	0.662857	0.331429	−	0.943480i	$-0.392469\pi$
$139$	92.1371	0.662857	0.331429	+	0.943480i	$0.392469\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	65.7266i	0.459627i
$144$	0	0
$145$	−72.1175	−0.497362
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.47496i	0.0233219i	0.999932	+	0.0116609i	$0.00371188\pi$
$149$	3.47496i	0.0233219i	−0.999932	+	0.0116609i	$0.996288\pi$
$150$	0	0
$151$	59.1323	0.391605	0.195802	−	0.980643i	$-0.437269\pi$
$151$	59.1323	0.391605	0.195802	+	0.980643i	$0.437269\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	392.999i	2.53547i
$156$	0	0
$157$	−290.527	−1.85049	−0.925244	−	0.379371i	$-0.876140\pi$
$157$	−290.527	−1.85049	−0.925244	+	0.379371i	$0.876140\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 195.289i	− 1.21297i
$162$	0	0
$163$	−272.752	−1.67332	−0.836662	−	0.547720i	$-0.815496\pi$
$163$	−272.752	−1.67332	−0.836662	+	0.547720i	$0.815496\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 188.581i	− 1.12923i	−0.825356	−	0.564613i	$-0.809026\pi$
$167$	− 188.581i	− 1.12923i	0.825356	−	0.564613i	$-0.190974\pi$
$168$	0	0
$169$	223.726	1.32382
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 88.4497i	− 0.511270i	−0.966773	−	0.255635i	$-0.917716\pi$
$173$	− 88.4497i	− 0.511270i	0.966773	−	0.255635i	$-0.0822845\pi$
$174$	0	0
$175$	−363.545	−2.07740
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3.06762i	0.0171375i	0.999963	+	0.00856877i	$0.00272756\pi$
$179$	3.06762i	0.0171375i	−0.999963	+	0.00856877i	$0.997272\pi$
$180$	0	0
$181$	12.4399	0.0687286	0.0343643	−	0.999409i	$-0.489059\pi$
$181$	12.4399	0.0687286	0.0343643	+	0.999409i	$0.489059\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	486.336i	2.62885i
$186$	0	0
$187$	102.332	0.547232
$188$	0	0
$189$	0	0
$190$	0	0
$191$	114.534i	0.599655i	0.953994	+	0.299827i	$0.0969290\pi$
$191$	114.534i	0.599655i	−0.953994	+	0.299827i	$0.903071\pi$
$192$	0	0
$193$	52.2218	0.270579	0.135290	−	0.990806i	$-0.456803\pi$
$193$	52.2218	0.270579	0.135290	+	0.990806i	$0.456803\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	237.428i	1.20522i	0.798037	+	0.602609i	$0.205872\pi$
$197$	237.428i	1.20522i	−0.798037	+	0.602609i	$0.794128\pi$
$198$	0	0
$199$	−30.7193	−0.154368	−0.0771841	−	0.997017i	$-0.524593\pi$
$199$	−30.7193	−0.154368	−0.0771841	+	0.997017i	$0.524593\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 69.6932i	− 0.343316i
$204$	0	0
$205$	105.632	0.515277
$206$	0	0
$207$	0	0
$208$	0	0
$209$	46.1875i	0.220993i
$210$	0	0
$211$	−9.25072	−0.0438423	−0.0219211	−	0.999760i	$-0.506978\pi$
$211$	−9.25072	−0.0438423	−0.0219211	+	0.999760i	$0.506978\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 545.565i	− 2.53751i
$216$	0	0
$217$	−379.787	−1.75017
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 611.451i	− 2.76674i
$222$	0	0
$223$	−209.727	−0.940482	−0.470241	−	0.882538i	$-0.655833\pi$
$223$	−209.727	−0.940482	−0.470241	+	0.882538i	$0.655833\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 197.712i	− 0.870978i	−0.900194	−	0.435489i	$-0.856575\pi$
$227$	− 197.712i	− 0.870978i	0.900194	−	0.435489i	$-0.143425\pi$
$228$	0	0
$229$	−365.590	−1.59646	−0.798230	−	0.602352i	$-0.794230\pi$
$229$	−365.590	−1.59646	−0.798230	+	0.602352i	$0.794230\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 261.121i	− 1.12069i	−0.828259	−	0.560345i	$-0.810668\pi$
$233$	− 261.121i	− 1.12069i	0.828259	−	0.560345i	$-0.189332\pi$
$234$	0	0
$235$	483.984	2.05951
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 203.712i	− 0.852350i	−0.904641	−	0.426175i	$-0.859861\pi$
$239$	− 203.712i	− 0.852350i	0.904641	−	0.426175i	$-0.140139\pi$
$240$	0	0
$241$	89.4943	0.371346	0.185673	−	0.982612i	$-0.440554\pi$
$241$	89.4943	0.371346	0.185673	+	0.982612i	$0.440554\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 136.741i	− 0.558128i
$246$	0	0
$247$	275.977	1.11732
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 487.339i	− 1.94159i	−0.239912	−	0.970795i	$-0.577118\pi$
$251$	− 487.339i	− 1.94159i	0.239912	−	0.970795i	$-0.422882\pi$
$252$	0	0
$253$	−80.1236	−0.316694
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 131.452i	− 0.511485i	−0.966745	−	0.255743i	$-0.917680\pi$
$257$	− 131.452i	− 0.511485i	0.966745	−	0.255743i	$-0.0823199\pi$
$258$	0	0
$259$	−469.987	−1.81462
$260$	0	0
$261$	0	0
$262$	0	0
$263$	265.186i	1.00831i	0.863613	+	0.504156i	$0.168196\pi$
$263$	265.186i	1.00831i	−0.863613	+	0.504156i	$0.831804\pi$
$264$	0	0
$265$	−492.083	−1.85692
$266$	0	0
$267$	0	0
$268$	0	0
$269$	366.503i	1.36246i	0.732068	+	0.681232i	$0.238555\pi$
$269$	366.503i	1.36246i	−0.732068	+	0.681232i	$0.761445\pi$
$270$	0	0
$271$	396.226	1.46209	0.731043	−	0.682331i	$-0.239034\pi$
$271$	396.226	1.46209	0.731043	+	0.682331i	$0.239034\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	149.156i	0.542387i
$276$	0	0
$277$	236.697	0.854500	0.427250	−	0.904133i	$-0.359482\pi$
$277$	236.697	0.854500	0.427250	+	0.904133i	$0.359482\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	383.899i	1.36619i	0.730331	+	0.683094i	$0.239366\pi$
$281$	383.899i	1.36619i	−0.730331	+	0.683094i	$0.760634\pi$
$282$	0	0
$283$	209.279	0.739500	0.369750	−	0.929131i	$-0.379443\pi$
$283$	209.279	0.739500	0.369750	+	0.929131i	$0.379443\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	102.081i	0.355682i
$288$	0	0
$289$	−662.991	−2.29409
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 352.123i	− 1.20179i	−0.799330	−	0.600893i	$-0.794812\pi$
$293$	− 352.123i	− 1.20179i	0.799330	−	0.600893i	$-0.205188\pi$
$294$	0	0
$295$	644.755	2.18561
$296$	0	0
$297$	0	0
$298$	0	0
$299$	478.750i	1.60117i
$300$	0	0
$301$	527.225	1.75158
$302$	0	0
$303$	0	0
$304$	0	0
$305$	597.856i	1.96018i
$306$	0	0
$307$	−286.542	−0.933362	−0.466681	−	0.884426i	$-0.654550\pi$
$307$	−286.542	−0.933362	−0.466681	+	0.884426i	$0.654550\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	328.791i	1.05721i	0.848869	+	0.528603i	$0.177284\pi$
$311$	328.791i	1.05721i	−0.848869	+	0.528603i	$0.822716\pi$
$312$	0	0
$313$	−39.2139	−0.125284	−0.0626420	−	0.998036i	$-0.519953\pi$
$313$	−39.2139	−0.125284	−0.0626420	+	0.998036i	$0.519953\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 144.471i	− 0.455744i	−0.973691	−	0.227872i	$-0.926823\pi$
$317$	− 144.471i	− 0.455744i	0.973691	−	0.227872i	$-0.0731768\pi$
$318$	0	0
$319$	−28.5939	−0.0896361
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 429.680i	− 1.33028i
$324$	0	0
$325$	891.231	2.74225
$326$	0	0
$327$	0	0
$328$	0	0
$329$	467.714i	1.42162i
$330$	0	0
$331$	156.879	0.473953	0.236977	−	0.971515i	$-0.423843\pi$
$331$	156.879	0.473953	0.236977	+	0.971515i	$0.423843\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 837.424i	− 2.49977i
$336$	0	0
$337$	−292.034	−0.866568	−0.433284	−	0.901257i	$-0.642645\pi$
$337$	−292.034	−0.866568	−0.433284	+	0.901257i	$0.642645\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	155.820i	0.456951i
$342$	0	0
$343$	−263.959	−0.769559
$344$	0	0
$345$	0	0
$346$	0	0
$347$	611.742i	1.76295i	0.472235	+	0.881473i	$0.343448\pi$
$347$	611.742i	1.76295i	−0.472235	+	0.881473i	$0.656552\pi$
$348$	0	0
$349$	348.789	0.999394	0.499697	−	0.866200i	$-0.333445\pi$
$349$	348.789	0.999394	0.499697	+	0.866200i	$0.333445\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 203.747i	− 0.577186i	−0.957452	−	0.288593i	$-0.906813\pi$
$353$	− 203.747i	− 0.577186i	0.957452	−	0.288593i	$-0.0931874\pi$
$354$	0	0
$355$	−206.108	−0.580586
$356$	0	0
$357$	0	0
$358$	0	0
$359$	476.892i	1.32839i	0.747560	+	0.664195i	$0.231225\pi$
$359$	476.892i	1.32839i	−0.747560	+	0.664195i	$0.768775\pi$
$360$	0	0
$361$	−167.065	−0.462783
$362$	0	0
$363$	0	0
$364$	0	0
$365$	321.907i	0.881937i
$366$	0	0
$367$	440.813	1.20112	0.600562	−	0.799578i	$-0.294943\pi$
$367$	440.813	1.20112	0.600562	+	0.799578i	$0.294943\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 475.541i	− 1.28178i
$372$	0	0
$373$	378.883	1.01577	0.507886	−	0.861424i	$-0.330427\pi$
$373$	378.883	1.01577	0.507886	+	0.861424i	$0.330427\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	170.853i	0.453191i
$378$	0	0
$379$	−518.418	−1.36786	−0.683929	−	0.729549i	$-0.739730\pi$
$379$	−518.418	−1.36786	−0.683929	+	0.729549i	$0.739730\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 143.249i	− 0.374019i	−0.982358	−	0.187009i	$-0.940121\pi$
$383$	− 143.249i	− 0.374019i	0.982358	−	0.187009i	$-0.0598795\pi$
$384$	0	0
$385$	−224.271	−0.582521
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 384.073i	− 0.987334i	−0.869651	−	0.493667i	$-0.835656\pi$
$389$	− 384.073i	− 0.987334i	0.869651	−	0.493667i	$-0.164344\pi$
$390$	0	0
$391$	745.385	1.90635
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 90.1547i	− 0.228240i
$396$	0	0
$397$	−10.6192	−0.0267486	−0.0133743	−	0.999911i	$-0.504257\pi$
$397$	−10.6192	−0.0267486	−0.0133743	+	0.999911i	$0.504257\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 399.582i	− 0.996463i	−0.867044	−	0.498232i	$-0.833983\pi$
$401$	− 399.582i	− 0.996463i	0.867044	−	0.498232i	$-0.166017\pi$
$402$	0	0
$403$	931.049	2.31030
$404$	0	0
$405$	0	0
$406$	0	0
$407$	192.828i	0.473779i
$408$	0	0
$409$	83.8067	0.204906	0.102453	−	0.994738i	$-0.467331\pi$
$409$	83.8067	0.204906	0.102453	+	0.994738i	$0.467331\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	623.080i	1.50867i
$414$	0	0
$415$	−422.291	−1.01757
$416$	0	0
$417$	0	0
$418$	0	0
$419$	650.655i	1.55288i	0.630193	+	0.776439i	$0.282976\pi$
$419$	650.655i	1.55288i	−0.630193	+	0.776439i	$0.717024\pi$
$420$	0	0
$421$	306.792	0.728721	0.364360	−	0.931258i	$-0.381288\pi$
$421$	306.792	0.728721	0.364360	+	0.931258i	$0.381288\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 1387.59i	− 3.26492i
$426$	0	0
$427$	−577.758	−1.35306
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 489.357i	− 1.13540i	−0.823236	−	0.567700i	$-0.807833\pi$
$431$	− 489.357i	− 1.13540i	0.823236	−	0.567700i	$-0.192167\pi$
$432$	0	0
$433$	55.1279	0.127316	0.0636581	−	0.997972i	$-0.479723\pi$
$433$	55.1279	0.127316	0.0636581	+	0.997972i	$0.479723\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	336.428i	0.769859i
$438$	0	0
$439$	−755.916	−1.72190	−0.860952	−	0.508687i	$-0.830131\pi$
$439$	−755.916	−1.72190	−0.860952	+	0.508687i	$0.830131\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	378.747i	0.854960i	0.904025	+	0.427480i	$0.140599\pi$
$443$	378.747i	0.854960i	−0.904025	+	0.427480i	$0.859401\pi$
$444$	0	0
$445$	500.302	1.12427
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 735.364i	− 1.63778i	−0.573948	−	0.818891i	$-0.694589\pi$
$449$	− 735.364i	− 1.63778i	0.573948	−	0.818891i	$-0.305411\pi$
$450$	0	0
$451$	41.8820	0.0928648
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1340.05i	2.94516i
$456$	0	0
$457$	−127.051	−0.278011	−0.139006	−	0.990292i	$-0.544391\pi$
$457$	−127.051	−0.278011	−0.139006	+	0.990292i	$0.544391\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 130.097i	− 0.282206i	−0.989995	−	0.141103i	$-0.954935\pi$
$461$	− 130.097i	− 0.282206i	0.989995	−	0.141103i	$-0.0450649\pi$
$462$	0	0
$463$	−349.830	−0.755572	−0.377786	−	0.925893i	$-0.623314\pi$
$463$	−349.830	−0.755572	−0.377786	+	0.925893i	$0.623314\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 165.590i	− 0.354582i	−0.984158	−	0.177291i	$-0.943267\pi$
$467$	− 165.590i	− 0.354582i	0.984158	−	0.177291i	$-0.0567334\pi$
$468$	0	0
$469$	809.273	1.72553
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 216.311i	− 0.457318i
$474$	0	0
$475$	626.287	1.31850
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 417.551i	− 0.871713i	−0.900016	−	0.435857i	$-0.856446\pi$
$479$	− 417.551i	− 0.871713i	0.900016	−	0.435857i	$-0.143554\pi$
$480$	0	0
$481$	1152.18	2.39537
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 373.631i	− 0.770373i
$486$	0	0
$487$	928.868	1.90733	0.953664	−	0.300875i	$-0.0972787\pi$
$487$	928.868	1.90733	0.953664	+	0.300875i	$0.0972787\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 515.364i	− 1.04962i	−0.851219	−	0.524811i	$-0.824136\pi$
$491$	− 515.364i	− 1.04962i	0.851219	−	0.524811i	$-0.175864\pi$
$492$	0	0
$493$	266.008	0.539569
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 199.180i	− 0.400764i
$498$	0	0
$499$	264.718	0.530497	0.265248	−	0.964180i	$-0.414546\pi$
$499$	264.718	0.530497	0.265248	+	0.964180i	$0.414546\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 381.801i	− 0.759048i	−0.925182	−	0.379524i	$-0.876088\pi$
$503$	− 381.801i	− 0.759048i	0.925182	−	0.379524i	$-0.123912\pi$
$504$	0	0
$505$	−871.779	−1.72630
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 111.470i	− 0.218998i	−0.993987	−	0.109499i	$-0.965075\pi$
$509$	− 111.470i	− 0.218998i	0.993987	−	0.109499i	$-0.0349246\pi$
$510$	0	0
$511$	−311.086	−0.608778
$512$	0	0
$513$	0	0
$514$	0	0
$515$	234.806i	0.455933i
$516$	0	0
$517$	191.895	0.371171
$518$	0	0
$519$	0	0
$520$	0	0
$521$	778.379i	1.49401i	0.664819	+	0.747005i	$0.268509\pi$
$521$	778.379i	1.49401i	−0.664819	+	0.747005i	$0.731491\pi$
$522$	0	0
$523$	−820.611	−1.56905	−0.784523	−	0.620100i	$-0.787092\pi$
$523$	−820.611	−1.56905	−0.784523	+	0.620100i	$0.787092\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1449.59i	− 2.75064i
$528$	0	0
$529$	−54.6170	−0.103246
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 250.251i	− 0.469514i
$534$	0	0
$535$	−117.191	−0.219049
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 54.2167i	− 0.100588i
$540$	0	0
$541$	75.9969	0.140475	0.0702374	−	0.997530i	$-0.477624\pi$
$541$	75.9969	0.140475	0.0702374	+	0.997530i	$0.477624\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	579.671i	1.06362i
$546$	0	0
$547$	11.9476	0.0218420	0.0109210	−	0.999940i	$-0.496524\pi$
$547$	11.9476	0.0218420	0.0109210	+	0.999940i	$0.496524\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	120.062i	0.217899i
$552$	0	0
$553$	87.1240	0.157548
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 272.715i	− 0.489614i	−0.969572	−	0.244807i	$-0.921275\pi$
$557$	− 272.715i	− 0.489614i	0.969572	−	0.244807i	$-0.0787246\pi$
$558$	0	0
$559$	−1292.49	−2.31215
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 908.020i	− 1.61282i	−0.591353	−	0.806412i	$-0.701406\pi$
$563$	− 908.020i	− 1.61282i	0.591353	−	0.806412i	$-0.298594\pi$
$564$	0	0
$565$	1.79558	0.00317801
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 761.146i	− 1.33769i	−0.743401	−	0.668846i	$-0.766789\pi$
$569$	− 761.146i	− 1.33769i	0.743401	−	0.668846i	$-0.233211\pi$
$570$	0	0
$571$	412.359	0.722169	0.361085	−	0.932533i	$-0.382407\pi$
$571$	412.359	0.722169	0.361085	+	0.932533i	$0.382407\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1086.45i	1.88948i
$576$	0	0
$577$	−65.3523	−0.113262	−0.0566311	−	0.998395i	$-0.518036\pi$
$577$	−65.3523	−0.113262	−0.0566311	+	0.998395i	$0.518036\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 408.095i	− 0.702401i
$582$	0	0
$583$	−195.107	−0.334660
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 782.957i	− 1.33383i	−0.745134	−	0.666914i	$-0.767615\pi$
$587$	− 782.957i	− 1.33383i	0.745134	−	0.666914i	$-0.232385\pi$
$588$	0	0
$589$	654.269	1.11081
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 422.167i	− 0.711917i	−0.934502	−	0.355958i	$-0.884155\pi$
$593$	− 422.167i	− 0.711917i	0.934502	−	0.355958i	$-0.115845\pi$
$594$	0	0
$595$	2086.37	3.50651
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 255.374i	− 0.426334i	−0.977016	−	0.213167i	$-0.931622\pi$
$599$	− 255.374i	− 0.426334i	0.977016	−	0.213167i	$-0.0683778\pi$
$600$	0	0
$601$	−1084.17	−1.80395	−0.901976	−	0.431787i	$-0.857883\pi$
$601$	−1084.17	−1.80395	−0.901976	+	0.431787i	$0.857883\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	92.0144i	0.152090i
$606$	0	0
$607$	533.012	0.878109	0.439054	−	0.898460i	$-0.355314\pi$
$607$	533.012	0.878109	0.439054	+	0.898460i	$0.355314\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 1146.60i	− 1.87660i
$612$	0	0
$613$	196.421	0.320426	0.160213	−	0.987082i	$-0.448782\pi$
$613$	196.421	0.320426	0.160213	+	0.987082i	$0.448782\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	274.696i	0.445212i	0.974908	+	0.222606i	$0.0714564\pi$
$617$	274.696i	0.445212i	−0.974908	+	0.222606i	$0.928544\pi$
$618$	0	0
$619$	−66.3032	−0.107113	−0.0535567	−	0.998565i	$-0.517056\pi$
$619$	−66.3032	−0.107113	−0.0535567	+	0.998565i	$0.517056\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	483.484i	0.776057i
$624$	0	0
$625$	273.201	0.437122
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 1793.87i	− 2.85193i
$630$	0	0
$631$	−33.3690	−0.0528828	−0.0264414	−	0.999650i	$-0.508418\pi$
$631$	−33.3690	−0.0528828	−0.0264414	+	0.999650i	$0.508418\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1320.72i	2.07987i
$636$	0	0
$637$	−323.953	−0.508560
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 173.753i	− 0.271066i	−0.990773	−	0.135533i	$-0.956725\pi$
$641$	− 173.753i	− 0.271066i	0.990773	−	0.135533i	$-0.0432747\pi$
$642$	0	0
$643$	−186.297	−0.289732	−0.144866	−	0.989451i	$-0.546275\pi$
$643$	−186.297	−0.289732	−0.144866	+	0.989451i	$0.546275\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 389.830i	− 0.602519i	−0.953542	−	0.301260i	$-0.902593\pi$
$647$	− 389.830i	− 0.602519i	0.953542	−	0.301260i	$-0.0974071\pi$
$648$	0	0
$649$	255.639	0.393897
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 825.230i	− 1.26375i	−0.775069	−	0.631876i	$-0.782285\pi$
$653$	− 825.230i	− 1.26375i	0.775069	−	0.631876i	$-0.217715\pi$
$654$	0	0
$655$	789.061	1.20467
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 889.786i	− 1.35021i	−0.737723	−	0.675103i	$-0.764099\pi$
$659$	− 889.786i	− 1.35021i	0.737723	−	0.675103i	$-0.235901\pi$
$660$	0	0
$661$	−1178.02	−1.78217	−0.891087	−	0.453832i	$-0.850057\pi$
$661$	−1178.02	−1.78217	−0.891087	+	0.453832i	$0.850057\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	941.682i	1.41606i
$666$	0	0
$667$	−208.277	−0.312259
$668$	0	0
$669$	0	0
$670$	0	0
$671$	237.044i	0.353270i
$672$	0	0
$673$	74.0154	0.109978	0.0549892	−	0.998487i	$-0.482488\pi$
$673$	74.0154	0.109978	0.0549892	+	0.998487i	$0.482488\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	747.194i	1.10368i	0.833949	+	0.551842i	$0.186075\pi$
$677$	747.194i	1.10368i	−0.833949	+	0.551842i	$0.813925\pi$
$678$	0	0
$679$	361.071	0.531768
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 522.059i	− 0.764362i	−0.924087	−	0.382181i	$-0.875173\pi$
$683$	− 522.059i	− 0.764362i	0.924087	−	0.382181i	$-0.124827\pi$
$684$	0	0
$685$	309.170	0.451344
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1165.79i	1.69200i
$690$	0	0
$691$	540.788	0.782617	0.391309	−	0.920260i	$-0.372023\pi$
$691$	540.788	0.782617	0.391309	+	0.920260i	$0.372023\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 770.722i	− 1.10895i
$696$	0	0
$697$	−389.626	−0.559004
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 733.604i	− 1.04651i	−0.852176	−	0.523256i	$-0.824717\pi$
$701$	− 733.604i	− 1.04651i	0.852176	−	0.523256i	$-0.175283\pi$
$702$	0	0
$703$	809.659	1.15172
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 842.473i	− 1.19162i
$708$	0	0
$709$	796.503	1.12342	0.561709	−	0.827335i	$-0.310144\pi$
$709$	796.503	1.12342	0.561709	+	0.827335i	$0.310144\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1134.99i	1.59185i
$714$	0	0
$715$	549.800	0.768950
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 38.3230i	− 0.0533004i	−0.999645	−	0.0266502i	$-0.991516\pi$
$719$	− 38.3230i	− 0.0533004i	0.999645	−	0.0266502i	$-0.00848403\pi$
$720$	0	0
$721$	−226.912	−0.314719
$722$	0	0
$723$	0	0
$724$	0	0
$725$	387.724i	0.534792i
$726$	0	0
$727$	277.210	0.381307	0.190654	−	0.981657i	$-0.438939\pi$
$727$	277.210	0.381307	0.190654	+	0.981657i	$0.438939\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2012.33i	2.75285i
$732$	0	0
$733$	−554.360	−0.756289	−0.378145	−	0.925747i	$-0.623438\pi$
$733$	−554.360	−0.756289	−0.378145	+	0.925747i	$0.623438\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 332.031i	− 0.450517i
$738$	0	0
$739$	−99.0236	−0.133997	−0.0669983	−	0.997753i	$-0.521342\pi$
$739$	−99.0236	−0.133997	−0.0669983	+	0.997753i	$0.521342\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1262.93i	1.69977i	0.526966	+	0.849886i	$0.323329\pi$
$743$	1262.93i	1.69977i	−0.526966	+	0.849886i	$0.676671\pi$
$744$	0	0
$745$	29.0678	0.0390172
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 113.252i	− 0.151204i
$750$	0	0
$751$	−831.309	−1.10694	−0.553468	−	0.832871i	$-0.686696\pi$
$751$	−831.309	−1.10694	−0.553468	+	0.832871i	$0.686696\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 494.638i	− 0.655150i
$756$	0	0
$757$	435.307	0.575042	0.287521	−	0.957774i	$-0.407169\pi$
$757$	435.307	0.575042	0.287521	+	0.957774i	$0.407169\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1419.98i	1.86594i	0.359959	+	0.932968i	$0.382791\pi$
$761$	1419.98i	1.86594i	−0.359959	+	0.932968i	$0.617209\pi$
$762$	0	0
$763$	−560.185	−0.734187
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1527.48i	− 1.99150i
$768$	0	0
$769$	−650.386	−0.845755	−0.422878	−	0.906187i	$-0.638980\pi$
$769$	−650.386	−0.845755	−0.422878	+	0.906187i	$0.638980\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	565.764i	0.731907i	0.930633	+	0.365953i	$0.119257\pi$
$773$	565.764i	0.731907i	−0.930633	+	0.365953i	$0.880743\pi$
$774$	0	0
$775$	2112.87	2.72629
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 175.857i	− 0.225747i
$780$	0	0
$781$	−81.7200	−0.104635
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2430.24i	3.09585i
$786$	0	0
$787$	350.192	0.444970	0.222485	−	0.974936i	$-0.428583\pi$
$787$	350.192	0.444970	0.222485	+	0.974936i	$0.428583\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.73522i	0.00219370i
$792$	0	0
$793$	1416.38	1.78610
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1424.25i	1.78702i	0.449044	+	0.893510i	$0.351765\pi$
$797$	1424.25i	1.78702i	−0.449044	+	0.893510i	$0.648235\pi$
$798$	0	0
$799$	−1785.19	−2.23428
$800$	0	0
$801$	0	0
$802$	0	0
$803$	127.633i	0.158945i
$804$	0	0
$805$	−1633.58	−2.02929
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 77.3959i	− 0.0956686i	−0.998855	−	0.0478343i	$-0.984768\pi$
$809$	− 77.3959i	− 0.0956686i	0.998855	−	0.0478343i	$-0.0152319\pi$
$810$	0	0
$811$	−61.4997	−0.0758320	−0.0379160	−	0.999281i	$-0.512072\pi$
$811$	−61.4997	−0.0758320	−0.0379160	+	0.999281i	$0.512072\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2281.55i	2.79945i
$816$	0	0
$817$	−908.263	−1.11170
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 589.940i	− 0.718562i	−0.933229	−	0.359281i	$-0.883022\pi$
$821$	− 589.940i	− 0.718562i	0.933229	−	0.359281i	$-0.116978\pi$
$822$	0	0
$823$	−393.084	−0.477623	−0.238811	−	0.971066i	$-0.576758\pi$
$823$	−393.084	−0.477623	−0.238811	+	0.971066i	$0.576758\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	848.031i	1.02543i	0.858559	+	0.512715i	$0.171360\pi$
$827$	848.031i	1.02543i	−0.858559	+	0.512715i	$0.828640\pi$
$828$	0	0
$829$	156.415	0.188680	0.0943398	−	0.995540i	$-0.469926\pi$
$829$	156.415	0.188680	0.0943398	+	0.995540i	$0.469926\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	504.375i	0.605492i
$834$	0	0
$835$	−1577.47	−1.88918
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1166.87i	− 1.39079i	−0.718628	−	0.695395i	$-0.755230\pi$
$839$	− 1166.87i	− 1.39079i	0.718628	−	0.695395i	$-0.244770\pi$
$840$	0	0
$841$	766.672	0.911619
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1871.46i	− 2.21474i
$846$	0	0
$847$	−88.9212	−0.104984
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1404.55i	1.65047i
$852$	0	0
$853$	−325.508	−0.381604	−0.190802	−	0.981629i	$-0.561109\pi$
$853$	−325.508	−0.381604	−0.190802	+	0.981629i	$0.561109\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1584.94i	− 1.84940i	−0.380693	−	0.924701i	$-0.624315\pi$
$857$	− 1584.94i	− 1.84940i	0.380693	−	0.924701i	$-0.375685\pi$
$858$	0	0
$859$	77.1665	0.0898329	0.0449165	−	0.998991i	$-0.485698\pi$
$859$	77.1665	0.0898329	0.0449165	+	0.998991i	$0.485698\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	290.286i	0.336369i	0.985756	+	0.168184i	$0.0537904\pi$
$863$	290.286i	0.336369i	−0.985756	+	0.168184i	$0.946210\pi$
$864$	0	0
$865$	−739.877	−0.855349
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 35.7455i	− 0.0411341i
$870$	0	0
$871$	−1983.93	−2.27776
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1350.53i	1.54346i
$876$	0	0
$877$	−1401.69	−1.59828	−0.799139	−	0.601146i	$-0.794711\pi$
$877$	−1401.69	−1.59828	−0.799139	+	0.601146i	$0.794711\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	242.922i	0.275734i	0.990451	+	0.137867i	$0.0440247\pi$
$881$	242.922i	0.275734i	−0.990451	+	0.137867i	$0.955975\pi$
$882$	0	0
$883$	503.337	0.570031	0.285015	−	0.958523i	$-0.408001\pi$
$883$	503.337	0.570031	0.285015	+	0.958523i	$0.408001\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	19.3501i	0.0218153i	0.999941	+	0.0109076i	$0.00347208\pi$
$887$	19.3501i	0.0218153i	−0.999941	+	0.0109076i	$0.996528\pi$
$888$	0	0
$889$	−1276.32	−1.43568
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 805.743i	− 0.902288i
$894$	0	0
$895$	25.6605	0.0286709
$896$	0	0
$897$	0	0
$898$	0	0
$899$	405.047i	0.450553i
$900$	0	0
$901$	1815.06	2.01450
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 104.059i	− 0.114982i
$906$	0	0
$907$	18.4644	0.0203577	0.0101788	−	0.999948i	$-0.496760\pi$
$907$	18.4644	0.0203577	0.0101788	+	0.999948i	$0.496760\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 210.717i	− 0.231303i	−0.993290	−	0.115652i	$-0.963104\pi$
$911$	− 210.717i	− 0.231303i	0.993290	−	0.115652i	$-0.0368956\pi$
$912$	0	0
$913$	−167.434	−0.183389
$914$	0	0
$915$	0	0
$916$	0	0
$917$	762.536i	0.831555i
$918$	0	0
$919$	1208.94	1.31549	0.657746	−	0.753240i	$-0.271510\pi$
$919$	1208.94	1.31549	0.657746	+	0.753240i	$0.271510\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	488.289i	0.529024i
$924$	0	0
$925$	2614.68	2.82668
$926$	0	0
$927$	0	0
$928$	0	0
$929$	592.318i	0.637586i	0.947824	+	0.318793i	$0.103278\pi$
$929$	592.318i	0.637586i	−0.947824	+	0.318793i	$0.896722\pi$
$930$	0	0
$931$	−227.649	−0.244521
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 856.004i	− 0.915512i
$936$	0	0
$937$	−1610.35	−1.71862	−0.859309	−	0.511456i	$-0.829106\pi$
$937$	−1610.35	−1.71862	−0.859309	+	0.511456i	$0.829106\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 334.091i	− 0.355038i	−0.984117	−	0.177519i	$-0.943193\pi$
$941$	− 334.091i	− 0.355038i	0.984117	−	0.177519i	$-0.0568072\pi$
$942$	0	0
$943$	305.067	0.323507
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1384.23i	− 1.46170i	−0.682538	−	0.730851i	$-0.739124\pi$
$947$	− 1384.23i	− 1.46170i	0.682538	−	0.730851i	$-0.260876\pi$
$948$	0	0
$949$	762.627	0.803611
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1378.46i	− 1.44645i	−0.690615	−	0.723223i	$-0.742660\pi$
$953$	− 1378.46i	− 1.44645i	0.690615	−	0.723223i	$-0.257340\pi$
$954$	0	0
$955$	958.071	1.00322
$956$	0	0
$957$	0	0
$958$	0	0
$959$	298.777i	0.311551i
$960$	0	0
$961$	1246.27	1.29685
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 436.833i	− 0.452676i
$966$	0	0
$967$	1296.49	1.34074	0.670368	−	0.742029i	$-0.266136\pi$
$967$	1296.49	1.34074	0.670368	+	0.742029i	$0.266136\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 955.174i	− 0.983701i	−0.870680	−	0.491850i	$-0.836321\pi$
$971$	− 955.174i	− 0.983701i	0.870680	−	0.491850i	$-0.163679\pi$
$972$	0	0
$973$	744.813	0.765481
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1295.70i	1.32620i	0.748529	+	0.663102i	$0.230761\pi$
$977$	1295.70i	1.32620i	−0.748529	+	0.663102i	$0.769239\pi$
$978$	0	0
$979$	198.365	0.202620
$980$	0	0
$981$	0	0
$982$	0	0
$983$	493.709i	0.502247i	0.967955	+	0.251123i	$0.0808000\pi$
$983$	493.709i	0.502247i	−0.967955	+	0.251123i	$0.919200\pi$
$984$	0	0
$985$	1986.07	2.01632
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1575.60i	− 1.59313i
$990$	0	0
$991$	−1079.44	−1.08924	−0.544620	−	0.838683i	$-0.683326\pi$
$991$	−1079.44	−1.08924	−0.544620	+	0.838683i	$0.683326\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	256.965i	0.258256i
$996$	0	0
$997$	−466.892	−0.468297	−0.234149	−	0.972201i	$-0.575230\pi$
$997$	−466.892	−0.468297	−0.234149	+	0.972201i	$0.575230\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1584.3.i.c.881.1		8
3.2	odd	2	inner	1584.3.i.c.881.8		8
4.3	odd	2		396.3.e.a.89.1	✓	8
12.11	even	2		396.3.e.a.89.8	yes	8
44.43	even	2		4356.3.e.f.485.1		8
132.131	odd	2		4356.3.e.f.485.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
396.3.e.a.89.1	✓	8	4.3	odd	2
396.3.e.a.89.8	yes	8	12.11	even	2
1584.3.i.c.881.1		8	1.1	even	1	trivial
1584.3.i.c.881.8		8	3.2	odd	2	inner
4356.3.e.f.485.1		8	44.43	even	2
4356.3.e.f.485.8		8	132.131	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1584.i (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-8.36495i q^{5} +8.08375 q^{7} -3.31662i q^{11} -19.8173 q^{13} +30.8543i q^{17} -13.9261 q^{19} -24.1582i q^{23} -44.9723 q^{25} -8.62139i q^{29} -46.9816 q^{31} -67.6201i q^{35} -58.1398 q^{37} +12.6279i q^{41} +65.2203 q^{43} +57.8586i q^{47} +16.3470 q^{49} -58.8268i q^{53} -27.7434 q^{55} +77.0781i q^{59} -71.4716 q^{61} +165.771i q^{65} +100.111 q^{67} -24.6395i q^{71} -38.4829 q^{73} -26.8108i q^{77} +10.7777 q^{79} -50.4834i q^{83} +258.095 q^{85} +59.8093i q^{89} -160.198 q^{91} +116.491i q^{95} +44.6662 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{7} - 8 q^{13} - 40 q^{19} - 48 q^{25} - 40 q^{31} - 56 q^{37} - 24 q^{43} + 96 q^{49} + 120 q^{61} + 272 q^{67} - 256 q^{73} - 32 q^{79} + 432 q^{85} + 96 q^{91} - 24 q^{97}+O(q^{100}) \) 8 * q + 16 * q^7 - 8 * q^13 - 40 * q^19 - 48 * q^25 - 40 * q^31 - 56 * q^37 - 24 * q^43 + 96 * q^49 + 120 * q^61 + 272 * q^67 - 256 * q^73 - 32 * q^79 + 432 * q^85 + 96 * q^91 - 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more